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A120612
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For n>1, a(n) = 2*a(n-1) + 15*a(n-2); a(0)=1, a(1)=1.
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5
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1, 1, 17, 49, 353, 1441, 8177, 37969, 198593, 966721, 4912337, 24325489, 122336033, 609554401, 3054149297, 15251614609, 76315468673, 381405156481, 1907542343057, 9536162033329, 47685459212513, 238413348924961
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Characteristic polynomial of matrix M = x^2 - 2x - 15. a(n)/a(n-1) tends to 5, largest eigenvalue of M and a root of the characteristic polynomial.
a(2n+1) = A005059(2n+1) = {1,49,1441,37969,966721,...} = (5^(2n+1) - 3^(2n+1))/2. a(2n) = A081186(2n) = {17,353,8177,198593,...} = (3^(2n) + 5^(2n))/2, 4th binomial transform of (1,0,1,0,1,......), A059841. - Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 31 2006
Binomial transform of [1, 0, 16, 0, 256, 0, 4096, 0, 65536, 0, ...]=: powers of 16 (A001025) with interpolated zeros . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 2008]
a(n) is the number of compositions of n when there are 1 type of 1 and 16 types of other natural numbers. [From Milan R. Janjic (agnus(AT)blic.net), Aug 13 2010]
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FORMULA
| Let M = the 2 X 2 matrix [1,4; 4,1], then a(n) = M^n * [1,0], left term.
a(n) = ( 5^n + (-1)^n * 3^n ) / 2. - Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 31 2006
a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*16^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 26 2007
If p[1]=1, and p[i]=16, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A [From Milan R. Janjic (agnus(AT)blic.net), Apr 29 2010]
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EXAMPLE
| a(4) = 353 = 2*49 + 15* 17 = 2*a(3) + 15*a(2).
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MATHEMATICA
| Table[(5^n+(-1)^n*3^n)/2, {n, 1, 30}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 31 2006
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CROSSREFS
| Cf. A005059, A081186, A059841.
Sequence in context: A146831 A146698 A146706 * A146461 A098329 A160076
Adjacent sequences: A120609 A120610 A120611 * A120613 A120614 A120615
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KEYWORD
| nonn
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 17 2006
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EXTENSIONS
| More terms from Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 31 2006
Entry revised. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 2008
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